# Maths basics

## Matrices

Matrices are rectangular arrays of numbers. In computer graphics, matrices are helpful for performing transformations in 2D or 3D space, like translation, rotation and scaling.

These transformations are done via matrix multiplication. The GLSL shader language has matrix multiplication built-in, in JavaScript you have to use an additional matrix library (or implement that yourself).

### Defining Matrices in GLSL

One mindtwisting detail when working with matrices in GLSL is, you don't write matrices in GLSL as you would write them down on paper, but in a transposed form:

``````mat3 identity() {
return mat3(
vec3(  1.,  0., 0.), // column 1
vec3(  0.,  1., 0.), // column 2
vec3(  0,   0 , 1.)  // column 3
);
}
``````

Above, you can see an example of the identity matrix, this behaves like the number `1`, when multiplying a matrix with the identity matrix, you will get the same matrix as the result (`A*I = A`).

### 2D transformation matrices

#### Rotation matrix

``````mat3 rotate2D(float a) {
float C = cos(a);
float S = sin(a);
return mat3(
vec3( C, S, 0.),
vec3(-S, C, 0.),
vec3( 0, 0, 1.)
);
}
``````

#### Translation matrix

``````mat3 translate2D(vec2 t) {
return mat3(
vec3(  1.,  0., 0.),
vec3(  0.,  1., 0.),
vec3( t.x, t.y, 1.)
);
}
``````

#### Scale matrix

``````mat3 scale2D(vec2 s) {
return mat3(
vec3( s.x,  0., 0.),
vec3(  0., s.y, 0.),
vec3(  0.,  0., 1.)
);
}
``````

#### Usage

``````const float PI = 3.141592654;
const float DEG = PI / 180.;

void main() {
mat3 mRotate = rotate2D(rotate * DEG);
mat3 mTranslate = translate2D(vec2(translateX, translateY));
mat3 mScale = scale2D(vec2(scaleX, scaleY));

// try changing the order of the multiplications :)
mat3 mTransform = mTranslate * mRotate * mScale;
vec3 pos = mTransform * vec3(position.xy, 1.);
vPos = pos.xy;
gl_Position = vec4(pos.xy,0.,1.);
}
``````

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### 3D transformation matrices

#### Rotation matrices

In 3D space, you can rotate around the X, Y and Z axis:

``````mat4 rotX(float angle) {
float S = sin(angle);
float C = cos(angle);
return mat4(
vec4(1.0, 0, 0, 0),
vec4(0  , C, S, 0),
vec4(0  ,-S, C, 0),
vec4(0  , 0, 0, 1.0)
);
}

mat4 rotY(float angle) {
float S = sin(angle);
float C = cos(angle);
return mat4(
vec4(C, 0  ,-S, 0),
vec4(0, 1.0, 0, 0),
vec4(S, 0  , C, 0),
vec4(0, 0  , 0, 1.0)
);
}

mat4 rotZ(float angle) {
float S = sin(angle);
float C = cos(angle);
return mat4(
vec4( C, S, 0  , 0),
vec4(-S, C, 0  , 0),
vec4( 0, 0, 1.0, 0),
vec4( 0, 0, 0  , 1.0)
);
}
``````

#### Scale matrix

``````mat4 scale(vec3 s) {
return mat4(
vec4(s.x, 0.0, 0.0, 0.0),
vec4(0.0, s.y, 0.0, 0.0),
vec4(0.0, 0.0, s.z, 0.0),
vec4(0.0, 0.0, 0.0, 1.0)
);
}
``````

#### Translation matrix

``````mat4 translate(vec3 p) {
return mat4(
vec4(1.0, 0.0, 0.0, 0.0),
vec4(0.0, 1.0, 0.0, 0.0),
vec4(0.0, 0.0, 1.0, 0.0),
vec4(p.x, p.y, p.z, 1.0)
);
}
``````

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